Incremental slope method for optimum continuous digital linearization

ABSTRACT

An incremental slope method for optimum continuous digital linearization is performed utilizing a programmed digital computer to determine the straight line segments best approximating a nonlinear function. By incrementing the slope of a straight line segment at each breakpoint on or near the curve of the nonlinear function, the difference between the value of the dependent variable on the function and the value of the dependent variable on the straight line segment for different slopes may be determined. By minimizing this difference and maximizing straight line segment length, the best approximating straight line segment is determined. In this manner, a linearized approximating function may be obtained for any set of data points stored in the computer corresponding to any nonlinear function such as a thermocouple curve. The straight line approximating segments with their corresponding preset values of slope and intercept are then utilized to linearize an analog-to-digital converter.

United States Patent i I 1 3,699,318

Underkoffler et al. 3 [451 Oct. 17, 1972 [54] INCREMENTAL SLOPE METHOD FOR gramming to Function Generation by Gracon &

OPTIMUM CONTINUOUS DIGITAL Strauss Simulation, November 1968. LINEARIZATION I Primary Examiner-Eugene G. Botz. [72] Inventors: Victor S. Underkoffler, Doylestown; I

I Richard B. white, Lansdale both of Attorney-Woodcock, Washburn, Kurtz &

I Pa Mackiewicz [73] Assignee: Leeds & Northrup Company, [57] ABSTRACT Philadelphia, Pa.

An incremental slope method for optimum continuous Filedl g- 1970 digital linearization is performed utilizing a pro- [211 Appl 64,977 grammed digital computer'to determine the straight line segments best approximating a nonlinear function. By incrementing the slope of a straight line segment at I each breakpoint on or near the curve of the nonlinear function, the difference between the value of the de pendent variable on the function and the value of the dependent variable on the straight line segment for [52] US. Cl ..235/l5l.l, 444/1, 340/347 AD {51] Int. Cl ..G06f 9/00 [58] Field of Search.. ......324/l32, 77 A; 235/15053;

v different slopes may be determined. By minimizing [56] References cued this difference and maximizing straight line segment UNITED STATES PATENTS length, the best approximating straight line segment is determined. In this manner, a linearized approximat- 3,513,30l 5/1970 Howe ..235/150.53

ing function may be obtained for any set of data points 3,557,347 1/l971 Robertson ..235/150.53 stored in the computer corresponding to any nonlinear OTHER PUBLICATIONS function such as a thermocouple curve. The straight line approximating segments with their corresponding preset values of slope and intercept are then utilized to linearize an analog-to-digital converter.

On the Approximation of Curves by Line Segments Using Dynamic Programming by Bellman, Communications of the ACM, Vol. 4, p. 284 June 1961.

Remarks on the Application of Dynamic Pro- 18 Claims, 13 Drawing Figures 22 THERMOCOUPLE ANALOG 1 INPUT VOLTAGE GATED CLOCK PULSES QS- REGISTER so, CLOCK PULSES VOLT -COUNT g? g CONVERSION CONVERTER END 5 5 P g is D. [I FULL SCALE COUNT 3 w 2 0 Q 1 I g I- 5 RUNUP LATCH, REGISTER RESET 0 z 9 AND DISPLAY LATCH z 2 u i 2 COUNT MODIFIER 8 k m m COUNT MODIFIER BREAKPOINT SEQUENCE IDENTIFIER BREAKPOINT IDENTIF ER CONTROL COUNT AND SIGNAL AND MODIFIER ENCQDER' DECODER ENABLING l Y PULSES 24 28 2s BREAKPOINT RESET PMENTEDIIN 17 I972 SHEET 1 BF 8 DISPLAYED VARIABLE TEMPERATURE NUMATRON (DIGITAL Ll NEARIZATION) THERMOCOUPLE TEMP.

MEASURED INPUT VARIABLE TEMPERATURE SEGMENT LENGTH SHEET 2 0F 8 l Jig- A ABCJD S LOPE LENGTH PATENTEDIIII mm I 3.699.318

SHEET 3 OF 8 SToRE POINTS /'3IO ON T FUNCTION cuRvE F lg. 4

DETERMINE INITIAL SEGMENT FROM To E 322 316 I DETERMINE SIGN OF -'3I4 BREAKPOINT DEv|ATIoN FROM cuRvE i I 330 I I I INCREMENTING I I WITH I COARSE DETERMINE NExT 3|6 STRAIGHT LINE I 'NCREMENTS I SEGMENT I I I I 332 I I I INCREMENT INCREMENTING INDEPENDENT I wITH I VARIABLE MEDIUM 2 I INCREMENTS I I 3l8 I Al I I DETERMINE SEGMENT I 334 I DEvIATIoN I FROM CURVE I INcREMENTING I I WITH I FINE 320 I I INCREMENTS I STORE LENGTH I I OF /7 SEGMENT 322 I F 79 5 INcREMENT SLOPE 324 DETERMINE SEGMENT HAVING M326 LONGEST LENGTH IDENTIFYING ,L- 328 CIRCUIT CONNECTION PATENTEDncI W 1912 SHEEI 5 OF 8 PATENTEDnm n 1972 SHEET 6 0F 3 h w g 0 a W i I Mm mam 3 1 1 E. E. whwwww m. Ev if fimmwlw W wm J n 1. L s) t, 1 W n n .w [a 3 I m u m 35 fig @g Q? u QM E man-1200mm 3.699.318 SHEET 8 BF 8 Fig. IO

.PLOTTER /364 WilllllllllllllllllllllllliIHIIHHII INCREMENTAL SLOPE METHOD FOR OPTIMUM CONTINUOUS DIGITAL LINEARIZATION BACKGROUND OF THE INVENTION 1. Field of the Invention This invention relates to the conversion of an independent variable input signal to a dependent variable output signal which is a nonlinear function of the input signal.

More particularly, this invention relates to a method for approximating the nonlinear function by a plurality of straight line segments.

The output of various sensors (voltage, current, resistance, etc.,) is, in general, a nonlinear function of the variable being measured (temperature, pressure, displacement, etc.). It is often desirable to obtain a direct reading in engineering units of the variable being measured from a signal obtained at the output of the various sensors. Various systems have been devised for obtaining such a reading. One such system is sold by Leeds & Northrup Company under the trademark NU- MATRON which provides, within the digital mode, linearization of these nonlinear sensor outputs by dynamically presetting both the sample period and the measuring period of a dual slope integrating digital voltmeter to predetermined values for each straight line approximating segment for the sensor curve. The method by which the straight line approximating segments are predetermined is of crucial importance for realizing the full efficiency of a system :such as the NU- MATRON and forms the basis of this invention.

2. The Prior Art Perhaps the most popular of prior art mathematical techniques for linearization is the Straight Line Approximating Method with Breakpoints on the Curve. The technique, utilizing a high speed digital computer, initiates at a first data point on the curve and extends straightline segments from this first or initial point through a second data point, a third, a fourth, etc. With each straight line approximation, the slope and intercept of the segment are calculated. Using the slope and intercept as calculated constants, at each value of the independent variable, the straight line approximating value of the dependent variable may be calculated. The straight line approximating value of the dependent variable is then subtracted from the true value of the dependent variable to yield the approximating error. The straight line approximating error at each value of the independent variable is then compared to a predetermined error or deviation limit. If the absolute value of the error at any value of the independent variable is less than the limit, a new straight line approximating segment is extended from the same initial point to a new final point which represents an increment along the curve. The calculation routine is now initiated for this new straight line approximating segment. That is all of the approximating error values on the new segment are determined and compared to the limit. The loop continues until any one of the absolute values of the approximating error in an interval is equal to or greater than the preselected initiating error limit. When this occurs, the loop is terminated and the segment is defined as a line between the initial point and the next to the last incremented point on the curve. The remaining approximating segments of the curve are calculated in the same manner initiating at the last point on the previous segment.

The straight line approximating method with the breakpoints on the curve is, in general, a fairly reliable method for linearization. Its greatest advantage is that each approximated segment is joined to the previous segment at the breakpoint. That is to say, the successive straight line approximations to the curve are completely continuous throughout the range of approximation. The greatest disadvantage being that the breakpoints must be on the curve as dictated by the method. This does not provide for optimum linearization as will be described later.

The best prior art way to approximate a set of discrete data points for the straight line is the classical method of Least Squares. The Least Squares method minimizes the sum of the squares of the differences from the true value to the approximating value. Adapting the Least Squares method to linearization of nonlinear sensor outputs, the technique includes the greatest possible number of data points (hence the longest segment) in a Least Squares straight line approximation within the limit of a preselected approximating error.

The greatest advantage of a successive Least Squares method of approximation is it provides for an optimum linearization. That is, for a given range and limit of error, the method fits the curve with the fewest number of segments. The greater disadvantage being that the method is inherently discontinuous at the breakpoints. Actually each segment is independent of the previous one. Depending on the nature of the error values at the breakpoint for adjacent segments, with the jump discontinuities that exist, it is possible to obtain the same approximating value for the dependent variables for discretely different independent variables on either side and close to the breakpoint. The opposite, and perhaps more serious case is also true. It is possible that no unique value of the independent variable will be obtained because of the jump discontinuity at the breakpoint. Essentially, a hole exists in the function near the breakpoint. This is particularly critical in a system such as the NUMATRON which is intended to display a dependent variable as a nonlinear function of an independent variable in the form of a nonlinear sensor output.

A dynamic programming approach attempting to minimize the maximum allowable difference between a nonlinear function and a straight line approximating segment at any given individual data point in that segment is disclosed in Remarks on the Application of Dynamic Programming to Function Generation, Gracon, T. J and Stauss, J. C., Simulation, Nov., 1968. This dynamic programming approach does not attempt to minimize the maximum allowable difference between the actual function and the straight line approximating segment at any given individual data point in that segment. The maximum difference between the actual function and the straight line approximation at a given data point could be different for each straight line approximating segment. Furthennore, this dynamic programming approach results in a discontinuous approximation which may be undesirable for the previously described reasons. Also, this dynamic programming approach appears to limit the minimum spacing between the break points.

In Automatic Techniques for Economic Computer Computation of Continuous Functions, Isadore L.

Goldhirsh, Simulation, July, 1968, various methods for method as described suffers from discontinuities in the approximating line segments.

A Secant approximation of a function is disclosed in US. Pat. No. 2,975,369 Vance. However, there is no method disclosed for achieving the optimum in a Secant approximation of a nonlinear function.

SUMMARY OF THE INVENTION In accordance with one aspect of the invention, a method is provided for obtaining an optimum, continuous straight'line segment approximation of a nonlinear function.

The method involves determining the difference between the value of one variable of the function and the approximated value of the one variable on a straight line segment for various increments of the other variable and various increments of slope for the straight line segment. The longest straight line segment which yields a difference within a predetermined range of deviation or error limit is then determined. The end of the straight line segment defines the next breakpoint on the function and the foregoing steps are repeated to determined the next approximating straight line segment.

In accordance with another aspect of the invention, the foregoing method is performed on a computing apparatus, each step of the method being executed by the program of the computing apparatus. The various values of the variables for the nonlinear function may be stored in the computing apparatus before the straight line segments are determined.

In accordance with another aspect of the invention, the determined straight line segments are utilized to identify appropriate circuit connections in a system for converting a sensor or transducer output signal to a linearized digital output. By utilizing the breakpoints, slopes and intercepts corresponding to the approximating straight line segments, appropriate electrical connections may be plotted or wired in response to the determined approximating linear segments.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 3 is a graphical illustration of one phase in the method of this invention;

FIG. 3a illustrates in further detail the aspect illustrated in FIG. 3;

FIG. 4 is a flow chart of one embodiment of the invention;

FIG. 5 is a flow chart of one block in the flow chart of FIG. 4;

FIG. 6 is a block diagram of a system constructed using the method of this invention;

FIG. 7 illustrates the relationship of the circuit diagrams in FIGS. 7a and 7b elaborating upon the block diagram of FIG. 6;

FIG. 8 is a timing diagram useful in explaining the operation of the block diagram of FIG. 6 and the circuit diagrams of FIGS. 7a and 7b;

FIG. 9 is a schematic illustration of the method for constructing the system disclosed in the block diagram of FIG. 6 and the circuit diagrams of FIGS. 7a and 7b; and

FIG. 10 is a schematic diagram also utilizing the invention in the manufacture of the system disclosed in the block diagram of FIG. 6 and the circuit diagrams of FIGS. 7a and 7b.

DETAILED DESCRIPTION OF THE INVENTION For ease in understanding the present invention, the method will first be described in terms of a specific application to a thermocouple as an exemplary type of transducer characterized by a nonlinear function. Accordingly, reference is now made to FIG. 1 wherein a nonlinear temperature-voltage function characteristic of a thermocouple is disclosed in conjunction with a system for displaying a measured variable temperature.

In FIG. 1, a curve 300 represents the conventional thermocouple temperature-voltage function with the measured variable temperature 301 at the input of a sensor or thermocouple transducer 302, plotted as the independent variable on a linear scale and the thermocouple transducer output voltage plotted as the dependent variable on a linear scale in terms of millivolts.

-As may be readily seen, the temperature-voltage function of the thermocouple is indeed nonlinear.

The same nonlinear relationship between temperature and voltage may be seen in a voltage-temperature curve 303 of FIG. 1 which may be thought of as the characteristic of a system 304 where the thermocouple output voltage is plotted as the independent variable system input on a linear abscissa scale and the temperature, which produces the thermocouple output voltage, is plotted as the dependent variable system output on a linear ordinate scale. Thus by coupling the thermocouple output voltage to the system 304 to be described hereinafter as the input to that system, a digital output may be obtained by the system 304 which is displayed as the temperature variable 305.

Digital linearization, in general, is concerned with the variables and parameters as outlined in the right hand portion of FIG. 1 with a thermocouple sensor serving as an example. The left hand portion of FIG. 1 relates to the input to the system 304.

Basically, the system 304 which may comprise a NU- MATRON numeric display system manufactured by Leeds and Northrup Company accepts the sensor analog output signal as the independent variable, performs the analog to digital conversion, performs linearization completely within the digital mode, then produces the dependent variable in displayed engineering units. The system 304 is primarily concerned with three basic parameters shown in FIG. 1: (l) the number of straight line approximating segments (S,,, (p 1, 2,. .K)), (2) the range of the function which is derstood to mean a discrete set of data to be approximated (x,, x,,) and (3) the maximum allowable approximating error (E) for an error or deviation at different values ofx, (x,-= 1,2. .n).

The first and foremost boundry condition of the system is the number of approximating segments. How the limited number of segments are used and where the breakpoints are placed (henceforth understood to mean the transition point x from one straight line approximating segment to another as x,- increases or decreases in value) is, as stated earlier, crucial to realizing full efficiency of the system. In general, however, using any of the common mathematical linearization techniques it can be understood, with the number of segments fixed at the maximum allowable dependent on the digital logic design of the system 304 that the range of approximation and the approximating error are interdependent. That is to say, by independently increasing (or decreasing) the range of the function to be approximated (x x,,) the absolute deviation or error limit E of the approximation increases (or decreases) given the boundry condition that the number of segments allowable is a predetermined integer. The inverse of the previous statement, with the absolute approximating error limit increasing (or decreasing) independently, is also true.

In the discussion to follow, the term curve is unpoints i yi O,l,2,. .n)) where the number of data points is very much greater than the number of straight line approximating segments, as is the case with thermocouple calibration tables. Or, as in the case of Spectray and Rayotube pyrometer, the term curve is defined by an analytic expression, y =f(x) which is used to generate the discrete set of points to be linearized For this example, the curve is understood to be monotonic over the range of approximation.

Referring to FIG. 2, the incremental slope method of this invention optimizes fit and continuity of the approximating straight line segments over the approximating range. The breakpoints B B B and B are not restricted to the curve 11 while at the same time continuity is maintained through the approximating segments 12, 13, and 14 characterized by slopes M M and M and'intercepts C C and C That is, the endpoint of the previous segment is the first point of the next segment. Initiation of the incremental slope method is completely dependent on the previous calculated segment. The incremented slope theorem reads as follows:

For a monotonic curve, if the approximation to the curve as calculated from the previous straight line approximating segment at its breakpoint is greater than the true value at that breakpoint, then the value of the slope of the next segment must be less than the previous or the approximation will diverge from the curve.

Conversely, if the approximation to the curve as calculated from the previous straight line approximating segment at its breakpoint is less than the true value at that breakpoint, then the value of the slope of the next segment must be greater than the previous or the approximation will diverge from the curve. Summarizing the theorem mathematically: mm independent variable at the breakpoint (ky =true dependent variable at a g =straight line approximating value atanm e approximating error at l'bp M value of slope for segment 8,,

Then: Mk M Conversely:

The incremental slope method as performed by a programmed digital computer will now be described with reference to FIG. 2 and the flow diagram of FIG. 4. Initially, the points on the curve 11 are stored in the computer as indicated at a block 310. As mentioned earlier, the incremental slope method is completely dependent on the previous segment. Although you can initiate the method by fixing a dummy segment as well be explained later, you can also determine the initial segment 12 using a Least Squares fit method described in the prior art section as indicated by a block 312.

The error at the breakpoint B2 is now checked for sign as indicated at a block 314. If the error e ispositive (or negative), then the value of the slope of the segment 13 starting at B which must be decreased (or increased) with respect to the slope M of the segment 12.

. Initially, the equation for a straight line segment terminating at breakpoint B and extending upwardly along the curve is determined as indicated at a block 316. The error or deviation e,- as shown in FIG. 1 is now determined for various values of x,- along the curve 11 and the determined straight line segment. This is accomplished in a loop including blocks 318 and 320 which increments and extends the determined segment by one data point (incrementing x,- by +1) until the absolute error of approximation becomes equal to or greater than the predetermined range of deviation designated as the limit error E. When the error e,- exceeds the value of the error limit E, the error analysis loop including the blocks 318 and 320 terminates, steps back one data point (incrementing x, by -l a counter records or stores the number of data points (x,,y,-) included in the approximating segment (hence the length of the segment for that particular incremented slope) within the error limit E as indicated at a block 322.

The slope of the determined segment terminating at the breakpoint B and extending upwardly along the curve 11 is now incremented from B as indicated at a block 324. By providing a slope incrementing loop including blocks 316-324, another straight line segment is determined including the determination of the intercept at block 316. Once again, the inner error analysis loop is repeated looking for all absolute e, values on the newest incremented straight line approximation until the error limit E is reached as x, increases. Also, a counter again records the maximum length of the new approximating segment within the error limit E. The slope incrementing loop continues incrementing the previous slope to determined the segment 13 and re-initiating the error analysis loop until the length of the approximating segment as recorded by the counter is less than the previous straight line approximation provided by the previous increment in slope. Here the slope incrementing loop terminates since the longest straight line approximation within the preselected initiating limit error E has been passed. At this time, the segment 13 has been determined mated by the straight line segments 12, 13, and 14.

Using the characteristic slopes M and intercepts C of the segments 12, 13, and 14, appropriate circuit connections for the system, 304 as shown in FIG. 1 and to be described in detail subsequently are identified. The circuit connections enable the system 304 to provide the digital linearization previously discussed. This step of identification is indicated at a block 328.

In the foregoing, a great deal of emphasis has been placed on the sign of the error at the breakpoints B and the direction of slope increment for the next line segment through that breakpoint. It becomes clear, in order to achieve a high degree of sensitivity, the value of the increment for the slope must be small so that the optimum fit (longest possible line segment) will be found. It is also clear, incrementing the slope in, very small bits'and running through the error analysis loop for each newly incremented slope in search of the optimum approximating segment would take a considerableamount of time even on a high-speed digital computer. Therefore, a system has been devised to implement the value of slope increment which will now be discussed with reference to FIG. 3 and the block diagram of FIG. 4.

The slope increments are taken in three steps,

I coarse, medium and fine in degree. The first slope increments (coarse increments) are 1 percent, for example, of the value of the slope of the previous segment corresponding to a block 330 in FIG. 5. Using 1 percent of the previous slopes value for the coarse increments moves the approximating segment to close proximity with the curve quickly regardless of the actual value of the slope of the previous segment. From FIG. 3, it is seen that coarse increment has produced a shorter approximating segment than coarse increment 4 within the limit E. It is also obvious that with the best coarse increment4, the optimum length approximating segment may have been passed over or not quite reached in the coarse incrementation of slopel-Iowever, it is certain that the slope of the optimum approximating segment lies somewhere about the slope yielded by coarse increment 4, that is, somewhere between the slope yielded by coarse increments 3 andS.

Now, the slope of the best coarse approximating segment (increment 4) is incremented by medium steps as indicated in a block 332 of FIG. 5 which are 0.1 percent or 0.l of the coarse increment size. The medium increments are made first in the negative direction from the best coarse approximation (medium increment A) and then in the positive direction from the best coarse approximation (medium increments A,B,C,C. The medium incremented slopes continue in ment within the limit E. That is, on the negative side medium incremented segment A is less than coarse incremented segment 4 so no further medium steps are taken in that direction and the calculation returns to the values yielded by coarse increment 4. On the positive side medium increment D is less than C so the calculation returns to the values yielded by medium increment C. The slope of the optimum length approximating segment has been confined to values around two medium incremented slopes. That is, in the range from coarse increment 4 to'medium increment A on the negative side or between medium increments B and D on the positive side. i

. The search is now'continued with fine increments within both these ranges as indicated by block 334 of FIG. 5. The value of fine-increment is 0.1 that of the medium increment (0.01 of the coarse increment, or 0.01 percent of the slope of the preceding segment). The first fine increment is a' on the negative side of the coarse increment 4. Since the finely incremented approximation segment yielded by slope increment a is shorter than the segment with slope yielded by coarse increment 4, the calculation again returns to the values yielded by coarse increment 4. In all cases, when fine increments are made from a coarseincrement with no intermediate medium increment, the search is made in only one direction (i.e., in the direction in which there are no medium increments) because in the opposite direction from the coarse increment, the search has already been made with medium increments (A,B,C and D from coarse increment 4 for example). The fine in:- crement search is now made on the positive side of coarse increment 4 as shown in FIG. 3a about the best medium increment segment C, first on the negative side of C with fine increment a whose approximating segment is shorter than that yielded by medium increment C, so the calculation returns to the latter. The positive side of medium-incremented segment C is now searched with fine segments a,b,c,d and e. Since the ap-' proximating segment yielded by fine increment e is shorter than that yielded by fine increment d, the calculations are terminated with the longest possible straight line approximation yielded by fine-incremented'slope d.

' The search is now complete on both the negative and positive sides of the approximating segment-yielded by the coarse increment-4. No better straight line approximating segments have been found on the negative side of 4 so the approximating segment with the slope yielded by coarse increment 4 is the best value found by the negative side search. However, on the positive side search with mediumand fine incremented slopes, the best value is that yielded by fine increment d. Now, the best approximation found in the negative side search is compared to the best approximation found in the positive side search. using the segment length criterion. It is now clear that the longest possible approximating segment within the error limit E is that yielded by finely incremented slope d, which is truly the optimum length approximating segment through the end point of the previous segment (z iy' All other incremented slopes have been eliminated as something less than the optimum approximating segment as now defined by the breakpoint, intercept and the slope yielded by fine increment d. I

Although the discussion of slope increments has been made with reference to curve 11, further analysis will prove that the three-step slope increment technique will hold for all monotonic curves. It should also be noted that using the three-step slope increment technique, after the best approximating segment has degree of sensitivity so that the optimum approximating segment is found.

. For simplicity of this discussion the Incremental Slope Method was initiated using a least squares approximation for the first segment. It should be appreciated that the first segment can also be calculated using the Incremental Slope Method by first setting a dummy segmenLThat is, since there is no value for slope and breakpoint error from the previous segment for incremental slope initiation, dummy values must be set. For example, if the monotonic curve to be linearized is an increasing (decreasing) function, set the dummy value for the breakpoint at the initial data point (x,,,y,,) and set the dummy value for slope at zero, then initiate the slope increments in the positive (negative) direction. One restriction for using the Incremental Slope Method on the first segment may be that the initiating point (x,,,y,,),is on the curve. Alternatively, the initial segment may be determined by incrementing with respect to the slope of the curve at an axis intercept. Incrementing with coarse increments will then proceed in one direction and then the other.

The discussion so far has shown how the Incremental Slope Method is unique for digital linearization applications. The method provides for optimum fit and continuity through the approximating segments. The'sensitivity of the methods approximation is limited only by the reliability of the input data to the computer. A general purpose computer program TCLIN will now be discussed which utilizes the Incremental Slope Method to linearize, for example, a thermocouple 7 curve. The program operator chooses the thermocouple to be linearized so that its respective data file is stored in the computer. The operator also chooses the range of approximation and the initiating limit of error E. Theprogram then begins to linearize the curve via the Incremental Slope technique. If more than 10 segments are required (the typical NUMATRON System maximum segment capability) the program returns CURVE CANNOT BE APPROXIMATED WITHIN SPECIFIED ERROR BY TEN SEGMENTS. The operator has the option of decreasing the range of the approximation or increasing the limit of error. If the original approximation takes less than ten segments, the operator then has a maximum segment option, whereby the computer automatically decreases the initiating limit of error by 0.l for example until all ten segments are used and the best approximation is made within the specified range using 100 percent of NU- MATRON systems capabilities.

The most critical values on the computer printout are preset 1 and preset 2, corresponding to slope and intercept of each approximating segment as well as pickoff, which indicates each segments position on the curve, i.e., its breakpoint. In actuality, these values represent the software logic design which is directly in- I corporated into a NUMATRON system hardware design via printed-circuit program cards to be described later.

TCLIN is a general conversational continuous linearization program designed to provide the best 10- segment approximation to a given data set. It is applicable to the complete set of thermocouple data files as 5 well as to any file of discrete data points. With minor modifications, TCLIN will. be applicable to analytic expressions such as polynomial equations as well as discrete data. While it was written specifically for application with a typical NUMATRON system, TCLIN is easily modified for general use.

The completely conversational nature of TCLIN makes it extremely straight-forward to use, even for persons with no previous programming experience.

PROGRAM DESCRIPTION A. OPTIONS To insure complete generality, TCLIN contains the following built-in options:

I. Input file variation-The user will be prompted for file name, at which time he may designate the data of his choice to be linearized.

2. Range variation-The user has the power to specify the temperature range over which the input data is to be linearized. In many cases, this range will be only a small subset of the data on file, so neither low end nor high end is fixed.

3. Variation of interval between data pointsAs the chosen range decreases, the accuracy provided by data at large intervals may, prove insufficient. On the other hand, as the chosen range grows, the extra accuracy provided by data at small intervals may prove meaningless. Therefore, provision has been made for the user to choose the density of his input data points. His choice is coarse approximation (every other data point: steps Centigrade or steps Fahrenheit) or fine approximation (every data point: 5 steps Centigrade or 10 steps Fahrenheit).

4. Automatic error adjustment-Depending upon which temperature range is chosen, the number of segments required to perform linearization within the specified error will vary. The user has the option to institute an automatic adjustment mechanism which reduces the allowed error until all 10 segments (or as many segments as possible) have been used. A listing of the final maximum error and the number of segments used is included in the output.

. Curtailed output-TCLIN will always list the number of segments and final maximum error for the total curve, as well as slope, intercept, span, pickoff, preset l and preset 2 for each segment. For immediate application, however, the user may not require a point-by-point comparison between the linearized data and the input data. This listing, therefore, can be deleted or directed onto a disk file for later printing B. ALGORITHMS Because of the requirement of continuity at the junction points between segments, two different algorithms have been used to calculate the straight line approximations.

1. Least Squares Approximation. This method provides the maximum length segment for any given error. 'That is, it includes the largest possible number of data points per segment by taking every point within the segment into account when calculating slope and intercept for that segment. Because the resulting line makesuse of these intermediate points, its begin-point cannot be set in advance. Thus, this algorithm can be used for the v first segment only.

Basically, the algorithm minimizes the sum of the squared differences between the true data and the calculated approximation.

v, 2. Incremental SlopeAlgorithm. This method takes advantage of the high-speed computers ability to perform a large number of iterative operations in a short time. In theory, the method is quite straightforward. The computer checks the endpoint of the segment last calculated. If the error at that point is positive (i.e., the calculated temperature is greater than the true temperature) then the slope of the new segment must be smaller than the slope of the last segment in order for the approximation not to diverge from the true curve. The opposite is true for a negative error at the endpoint.

Having decided on the sign of the error, the computer increments the slope of the new segment in the proper direction, and, using the new slope and the endpoint of the previous segment, calculates a new intercept. It then calculates as many points as possible until the error at any point becomes too large, at which point it saves the number of data points used. This procedure is repeated until the number of points used decreases, indicating that the optimal slope has been passed. It then steps back one increment and searches with increments one-tenth and finally one-one-hundredth the original size, thereby pin-pointing the best slope.

. Maximization of number of segments (optional). Before printing the final results, the program checks whether all ten segments have been used. If

so, it goes ahead with the printing. If not, it lowers the allowed error by 0. 1 and sets the indicator FLAGZ equal to l to show that segment max- 7 imization is in progress. This process is repeated until ten or more segments are used.

When the number of segments needed becomes greater than ten, allowable error is increased by 0. 1, insuring that the number of segments needed will bethe maximum less than or equal to 10. The indicator FLAG3 is set to 1, showing that no further error incrementation is needed. FLAG3 is also set to 1 if the user does not opt for maximum number of segments, since if F LAG3 equals 1 the printing will proceed no matter how many segments have been used.

If more than ten segments have been used and FLAGZ is 0, then the curve cannot be approximated in 10 segments within the original error, and this information is printed. If, however, more than ten segments are needed and FLAGZ is 1, then the program has merely incremented error once too often. This is easily corrected by adding one increan absolute error check is the basis for determining the ment to "error and noting that the program should terminate with the resulting set of segments.

C. STRUCTURE six subroutines are:

1. INPUT: This routine conversationally allows the user to perform the above options as well as initialize certain input parameters. The conversational nature of the routine makes the program extremely straightforward to operate.

FILTYP: Called by INPUT, this routine decides which data file has been chosen and returns its characteristics for printing by INPUT. READIN: Provided with the name of the file to be processed and the range to be linearized within that file, READIN retrieves the proper data from storage and prepares it for use. This preparation includes shifting the data points until the point with subscript 0 has the value 0. This shift is made necessary by the generality of the program, which must accept blocks of data much smaller than the maximum variable dimension and which are, therefore, misaligned with respect to subscript. LSTSQRi This routine calculates the first segment using the least squares I criterion as described above. It returns values for slope, intercept, segment length and segment end-points." I NUSLOP: The basis of the: Incremental Slope Algorithm is contained in this routine, which provides an updated value for segment slope and intercept based upon the old value and the present state of the algorithm'Slope is updated by adding successively smaller increments to an'initial estimate. Increment size is reduced each time incrementation by the previous size produces a decline in segment length. NUSLOP features the ability to store the final values obtained by the large and medium size steps and to I search values both slightly lower and slightly higher to obtain optimal results. Bookkeeping is done by the counter FLAG which keeps track of how many times the routine has been called, and therefore of what value of increment should be used. 1 6. QUIT: This is an output routine which tabulates segments, breakpoints, preset values and segment lengths. Depending upon the point-list option described above, it also tabulates true temperature, thermocouple EMF, amplified EMF, approximate temperature and error for each data point either at the terminal or onto a disk file called PT- LIST for later printing. If the latter option is chosen, typing SCHED(PRINT) at the termination of the program will cause the disk file to be printed at a high-speed printer. For this option, approximately 1 minute per should be allowed for program termination. D. ERROR CRITERION Regardless of which algorithm is being implemented,

length of a segment. Using the calculated slope and intercept, the computer applies the straight-line formula to each independent variable value (millivolts) and determines an approximate dependent variable value (temperature). If the absolute difference between the latter and the true temperature at any point within the segment is greater than a predetermined amount, the segment has included one point too many and a final segment is calculated using one point fewer. I

The following is a list of variables appearing in TCLIN and their definitions. Variables used in 13 subroutines only will be preceded by an Subscripted variables (arrays) will be followed by their range of subscripts.

*8 (appears in LSTSQR) sum of independent variable values for least squared segment calculation.

BP() number of counts at begin-point of each segment: pickoff.

*C (appears in LSTSQR) sum of dependent variable values for least squared segment calculation.

*CI-IARI (appears in QUIT) character string representation of my output of sensor.

*CI-IAR2 (appears in QUIT) character string representation of my output of pre-amplifier.

*CI-IAR3 (appears in QUIT) character string representation of linearized approximation.

*CHAR4 (appears "in QUIT character string representation of error at each data point.

C1(10) preset 2 for each segment; intercept or complement of intercept, whichever is positive.

*D (appears in LSTSQR) sum of independent variable values multiplied by dependent variable values for least squared segment calculation.

*DELT (appears in QUIT) denominator (determinant of Cramers rule matrix: E* N+1 )B used in least squared segment calculation.

DIFF(38:353) deviation of approximation from true curve at each point.

*E (appears in LSTSQR) sum of independent variable values squared for least squared segment calculation.

END(10) endpoint independent variable value of each segment.

ERROR maximum allowable deviation from true curve; chosen by user.

FLAG indicator: used in LSTSQR to indicate the ERROR has been exceeded, and in NUSLOP to record status of INC.

FLAGl indicator used to show that INC has taken on every required value and N USLOP need not be recalled (NUSLOP will be called unless FLAG1=0). FLAGZ indicator used to show that maximum K index of endpoint of segment N (or begin-point of segment N+l) during calculation of segment N+l.

K2 counter storing largest number of data points used in each Incremental Slope calculation.

L index showing segment number; final value indicates total number of segments used.

*LINE( 121) (appears in QUIT) character string of length 121 containing CHARl through CHAR4 to be written on disk file.

*LO (appears in READIN) index of Y containing data point 0.

LOW low end of temperature range to be linearized.

*MAX (appears in INPUT and FILTYP) indicates high end of maximum range of curve to be linearized.

*MIN (appears in INPUT and FILTYP) indicates low end of maximum range of curve to be linearized.

N number of points (excluding begin-point) in each segment;incremented during segment calculation.

N1 complement of slope for each segment; preset *NAME (appears in INPUT and FILTYP) name of chosen data file.

*NMED (appears in NUSLOP) length of best segment in Incremental Slope Algorithm after medium value of positive INC.

*NNEG (appears in NUSLOP) length of resulting segment in Incremental Slop Algorithm after medium value of negative INC.

*NPOS (appears in NUSLOP) length of best segment in Incremental Slope Algorithm after smallest value of INC in original direction.

*NSM (appears in NUSLOP) length of best segment in, Incremental Slope Algorithm after smallest value of INC.

NUM maximum number of segments option indicator.

PTS point-by-point list option indicator.

REPLY dummy input character string.

*SKIP (appears in READIN) indicator showing whether every data point or every other data point is to be used.

SLOPE( l0) slope of each segment.

*SLOPEl (appears in NUSLOP) value of slope of best segment in Incremental Slope Algorithm after smallest value of INC in original direction *SLOPE2 (appears in NUSLOP) value of slope of best segment in Incremental Slope Algorithm after medium value of INC.

*SLOPE3 (appears in NUSLOP) value of slope in Incremental Slope Algorithm after largest value of INC.

SPAN(10) length of each segment in engineering units.

STEP interval between data points; numerical representation.

T(38:353) dependent variable (temperature);

true curve.

*TITLE (appears in INPUT and FILTYP) title of chosen data file.

TURN dummy input character string.

TYPE temperature scale of chosen data file (Fahrenheit or Centigrade).

X(38:353) independent variable (mv).

*Y(38:353) (appears in READIN) dummy array. containing data points; processed and transferred to X.

YCALC(38:353) approximation to dependent variable; linearized curve.

YCEPT(10) intercept for each segment.

*YCEPTI (appears in NUSLO P) intercept of best segment in Incremental Slope Algorithm after smallest value of INC in original direction.

*YCEPT 2 (appears in NUSLOP) intercept of best segment in 'lncremental 810 medium value of INC.

*YCEPT3 (appears in NUSLOP) intercept in In- START:

SEE:

SEGMENT FOUR 2 cremental Slope Algorithm'after largest value of INC.

The following sets forth the numbered instructions in the main program followed by the instructions of the P Algorithm t r 5 subroutines. This program and the subroutines may be performed by accessing a computer at an IBM 2741 terminal.

TCLIN DECLARE X(.-38:353) DEC(6), sLoPE(1)' DEC(A6),, YCEPT(1) DEC(6), BP(1) DEC(6); v 1 DECLARE INPUT ENTRY EXT;

DECLARE NUSLOP ENTRY EXT; DECLARE Ls s R ENTRY EXT, READIN ENTRY EXT, ENTRY EX 1 DECLARE REPLAY CHAR(2) VAR;

' CALL READIN (REPLY, LOW,HIGH', STEP, TYPE,X);

FRE READIN;

CALL LSTSQR(X, T, N, L, H, K, ERROR, FLAG, SLOPE, YCEPT,

,YCALC,DIEE);

IMAGE;

SLOPE INTERCEPT LOOPZ:

ONE:

TWO:

BIG:

SEVEN:

MANY:

TERM:

OOPS:

QUIT:

STOP:

TCLIN TO MANY;

FREE LsTsQR;

IF DIFF(K) THEN INc=.1*sLOPE(L); ELSE INC:

IF Lvl THEN GO To MANY;

sLOPE(L)=sLOPE(L-1);

SLOPE(L)=SLOPE(L)+INC;

IF ABS(DIFF(M) )-:'ERROR THEN GO TO GO;

IF K27N THEN GO TO BIG;

GO TO ONE;

IF K+N]-;-H THEN GO To Two;

GO TO FOUR;

IF FLAG= THEN-FLAO=I;

CALL NUsLoP(sLOPE, INC, YCEPT, END(L-l), SLOPE(L-1), YCEPT(L-l),FLAG,K2, FLAG1,L); IF FLAGlI-a 5 THEN GO To ONE;

no I=K TO K+N; YCALC(I)=SLOPE(L)*X(I )+YCEPT(L); DIFF(I)=YCALC(I)-T(I);

ELSE GO TO SEVEN;

- IF ERROR .5 THEN FLAG3=1;

FLAG2=15 GO TO START; ERROR=ERROR+.1; FLAG3=1;

GO TO START;

- PUT LIST( CURVE CANNOT BE APPROXIMATED WITHIN SPECIFIED ERROR BY l SEGMENTS' PUT LIST( FREE NUSLOP;

CALL QUIT(L SPAN,BP,N1.C1,T, X, YCALC,DIFF, LOW, STEP, PTS, ERROR, NUM, GAIN);

END;

, INPUT 1 INPUT: PROCEDURE (REPLY, Low, HIGH, STEP,TYPE, ERROR, PTs,

: NUM,GAIN); 2'. DECLARE TITLE cRAR(2) vAR, INTvAL CHAR(2); -3.' DECLARE NAME GHAR(2) VA 1. DECLARE FILTYP ENTRY EXT; 5.. PUT LIsT( ENTER NAME OF DATA FILE To BE APPROXIMA'IED' 6. FILIN: READ INTO(REPLY); 3

7'. GALL FILTYP(REPLY,TYPE,MAX,MIN,TITLE); 8. PUT IMAGE(TITLE, TYPE,MIN,MAX,TYPE) (F);

9. F: IMAGE;

-- -.--cU vE IN DEGREES MAXIMUM RANGE IS To --.----'---DEGREES. 1 FREE FILTYP; 11. PUT LISTY 12. PUT. LIsT( 'ENTER THE DESIRED INTERVAL BETWEEN DATA r POINTS 13. PUT LIST( YOUR CHOICE IS 5- 0R: 1-DEGREE STEPS CENTIGRADE');

1M. PUT L sT( OR 1- 0R 2-DEGREE STEPS FAHRENHEI'I" 15. READ INT0(INTvAL); 16. IF INTVAL='5' THEN INTvAL= '5'; 17. TEP='FLoAT(INTvAL); 18. PUT LIST( 19. PUT LIST( ENTER vALUEs FoR Low AND HIGH E DS 0 RANGE DESIRED 2. PUT IMAGE(sTE P TYPE) (G); 21. G: IMAGE; VALUES FoR Low AND HIGH ENDS MUsT BE MULTIPIEs 0F ------DEGRE s 22. PUT ,LIsT( 23. T LIST(LOW,HIGH); .49 25. PUT LIsT( ENTER MAXIMUM ALLOWABLE ERROR 26. P T LIs 27. GET IsT(ERRoR); 28. PUT LIST( 29. PUT LIST( ENTER PRE-AMPLIFIER GAIN 3 PUT LIsT( 31. GET LIST(GAIN); 32. I PUT LIsT( )g I 33- PUT LIST( 'ENTER 1 .IF IMMEDIATE POINT BY POINT LIST Is REQUIRED,'); -3 PUT LIsT( '2, IF LIsT Is To BE WRITTEN ON FILE FOR LATER PRINTING, 35. A PUT LIST( 0R .-'IF N0 LIsT Is DEsIRED' 36. READ INTO(PTS); I 37. PUT LIsT( A 38. PUT LIsT( 'ENTER 1 FoR MAXIMUM NUMBER OF sEGMENTs,

, FoR FIXED. ERRoR' A 39- READ INT0(NU M) A PUT LIsT( +1. ETURN;

' END INPUT;

3,699,318 21 22 FILTYP PROCEDURE (REPLX, TYPE,MAX,MIN,TITLE);

PLY=uPc ;,sE(R PLY),;' f

'I-F SUB STR(REPLY, 1, 1)= 'c' THEN TYPE= 'CENTIGRADE' ELsE TYPE='FARRENHEIT' IF REPLY: 'CTYPJ' THEN MAX=87;

IF REPLY='CTYPJ THEN MIN=-l9;

IF REPLY: CTYPJ' .'REPLY= 'FTYPJ' THEN TITLE='IRON coN- IF REPLY: 'FTYPJ' THEN MIN=-31'; sTANTAN' IF REPLY: 'FTYPJ' THEN MAx=16;

IF REPLY="FTYPK' THEN MAx=2 +9;

IF REPLY= 'FTYPK' :RE Ly='cTYPK' THEN MIN=;

IF REPLY: 'FTYPK' .REPLY: 'CTYPK' THEN TITLE='CHROMEL ALUMEL' IF REPLY: 'CTYPK' THEN MAX=1375 IF REPLY: CTYPT' :REPLY= 'FIYPT' 1 THEN TITLE= COPPER CONSTANTAN'; 4

IF REPLY: 'CIYPT' THEN MAX=395;

IF REPLY: 'CTYPT' THEN MIN=-'19;

IF REPLY: 'FTYPT' THEN MAX=75;

IF REPLY: 'FTYPT THEN NIN=-3I;

IF REPLY: 'CTYPS .'REPLY= 'FTYPS' THEN TITLE= 'PLAT+ 10%R-HOD v PLAT' REPLY: CTYPS' THEN MAX=1765;

IF REPLY='CTYPS',!REPLY='FTYPS' !REPLY='CTYPR' I REPLY: 'FTY PR THEN MIN=;

IF REPLY: 'CTYPR' !REPLY=,'FTYPR' THEN TITLE= 'PLAT+l3%R-HOD v PLAT IF REPLY: 'CTYPR THEN MAx=1695;

IF REPLY: 'FTYPR' THEN -MAX=39;

IF REPLY='CTYPE' .'REPLY=.'FTXPE'THEN TITLE: "CHROMEL CONSTANTAN';

IF REPLY= 'CTYPE THEN MAX=1;

IF REPLY='CTYPE' THEN MIN=-2;

IF REPLY='FTYPE' THEN MAX=183;

IF REPLY: 'FTYPE' THEN MIN=-32;

IF TITLE: NONE THEN PUT LIsT( FILE' REPLY, NOT AVAILABLE, PLEAsE REENTER' IF TITLE: 'NONE' THEN READ INTO(REPLY) ELSE RETURN; GO TO UP;

RETURN;-

END FILTYP; 

1. In the manufacture of a system for generating a dependent output variable which is a nonlinear function of an independent input variable, said function capable of approximation by a plurality of linear segments joining breakpoints near the curve representing said nonlinear function, each of the linear segments having characteristic slopes and intercepts, the method comprising the following step:
 1. storing the values of variables corresponding to a plurality of points on said curve representing said nonlinear function in a programmed, automatic computing apparatus; the method also comprising the following programmed steps executed by said computing apparatus:
 2. determining the difference between the value of one of the function variables on said curve and the approximated value of said one variable on a straight line segment having a particular slope and intercept for a particular value of the other of the function variables;
 3. incrementing said other variable and repeating step 2 so long as the difference obtained from step 2 lies within a predetermined range of deviation thereby incrementing the length of the straight line segment;
 4. incrementing the slope of the straight line segment when the difference of step 2 exceeds a predetermined range of deviation;
 5. storing the length of the straight line segment before the slope is incremented;
 6. determining the slope and intercept characteristic of the straight line segment having the longest length; the method further comprising the step of
 7. identifying a circuit connection for said system corresponding to the straight line segment having the longest length.
 2. determining the difference between the value of one of the function variables on said curve and the approximated value of said one variable on a straight line segment having a particular slope and intercept for a particular value of the other of the function variables;
 2. determining the maximum length for one straight line segment in said set such that said difference does not exceed a predetermined range of deviation and the other end terminates off the curve;
 2. The method of claim 1 wherein one end of said line segment pivots about the breakpoint while the slope of the straight line segment is incremented and the value of said other variable is incremented.
 2. incrementing said other variable and repeating step 1 so long as the difference obtained from step 1 lies within a predetermined range of deviation thereby incrementing the length of the straight line segment;
 2. repeatedly applying a scanning function with respect to said straight line segment to identify a plurality of straight line segments having different lengths and extending from said breakpoint in different angular directions with respect to said one terminal breakpoint;
 3. identifying the one of said straight line segments representing a line having the maximum length, originating at said breakpoint, and terminating at its opposite end at another breakpoint near said curve, the deviation of said straight line throughout its length from said curve not exceeding said predetermined limit, repeating the foregoing steps utilizing each last-named breakpoint for scanning until there has been identified that plurality of straight line segments coextensive with and approximating said curve and none of which throughout the respective lengths thereof deviates from said curve beyond said predetermined limit; determining the slopes and intercepts of each of said straight lines approximating said curve; and applying said slopes and intercepts to a system for generation of said linearized digital output for said dual slope analog-to-digital converter.
 3. incrementing the slope of the straight line segment when the difference of step 1 exceeds a predetermined range of deviation;
 3. The method of claim 2 wherein steps 1-7 are repeated using the other end of said straight line segment having the longest length as the next breakpoint until all of said plurality of segments are determined.
 3. repeating the foregoing steps utilizing the other end of said one straight line segment as another breakpoint; and
 3. incrementing said other variable and repeating step 2 so long as the difference obtained from step 2 lies within a predetermined range of deviation thereby incrementing the length of the straight line segment;
 4. incrementing the slope of the straight line segment when the difference of step 2 exceeds a predetermined range of deviation;
 4. determining slopes and intercepts on the dependent variable coordinate axis for each determined straight line segment joining breakpoints on or near said curve.
 4. The method of claim 3 wherein the incrementing of the slope in step 4 continues until the length of the straight line segment characterized by the incremented slope begins to decrease.
 4. storing the length of the straight line segment before the slope is incremented; and
 5. determining the increment of slope and intercept characteristic of the straight line segment having the longest length.
 5. The method of claim 4 wherein the increment of the slope characteristic of the straight line segment having the longest length is determined in step 6 by counting the number of incremented values of said other variable for each straight line segment characterized by a particular slope increment.
 5. storing the length of the straight line segment before the slope is incremented;
 6. determining the slope and intercept characteristic of the straight line segment having the longest length; the method further comprising the step of
 6. The method of claim 5 including the step of computing each straight line segment terminating at each breakpoint for each characteristic increment of slope.
 7. The method of claim 4 wherein the incrementing of the slope in step 4 comprises the steps of incrementing the slope with coarser or larger increments initially so as to roughly determine the characteristic slope of the straight line segment having the longest length and incrementing the slope with finer or smaller increments subsequently in the vicinity of the roughly determined characteristic slope to more precisely determine the characteristic slope of the straight line segment having the longest length.
 7. identifying a circuit connection for said system corresponding to the straight line segment having the longest length.
 8. The method of claim 7 wherein the slope is increased and decreased by incrementing with the finer or smaller increment in the vicinity of the roughly determined characteristic slope of the line segment.
 9. The method of claim 8 wherein the slope is incremented utilizing increments having at least three degrees of coarseness, said slope being incremented with a finer increment on both sides of a coarser increment to determine the appropriate direction of incrementation to locate the straight line segment having the longest length.
 10. The method of claim 4 wherein the identification of the circuit connection of step 7 comprises the marking of the connection in said system.
 11. The method of claim 4 wherein the identification of the circuit connection in step 7 comprises the wiring of the circuit connection in said system.
 12. In the manufacture of a system including a analog-to-digital converter for generating a linearized digital output as a dependent variable which is a nonlinear function of an input as an independent variable, said function being approximated by a plurality of continuous linear segments joining breakpoints on or near a curve plotted with respect to a pair of Cartesian coordinate axes and representing said nonlinear function, the method including the following steps each executed by a programmed, automatic computing apparatus:
 13. The method of claim 12 further including the step of identifying circuit connections for said system in response to and corresponding to said determined slopes and intercepts.
 14. The method of approximating a nonlinear function by a plurality of straight line segments joining breakpoints near a curve representing said nonlinear function, the method comprising the following programmed steps executed by a programmed computing apparatus:
 15. The method of claim 14 wherein one end of said line segment pivots about the breakpoint while the slope of the straight line segment is incremented and the value of said other variable is incremented.
 16. The method of claim 15 wherein steps 1-7 are repeated using the other end of said straight line segment having the longest length as the next breakpoint until all of said plurality of segments are determined.
 17. The method of claim 16 wherein the slope is incremented in a direction so as to increase or decrease in a direction determined by the sign of said difference at the breakpoint of the previous straight line segment.
 18. A method of establishing a linearized digital output from a dual slope analog-to-digital converter approximating a nonlinear function characterized by a curve having an independent variable plotted along one Cartesian coordinate axis and a dependent variable plotted along the other coordinate axis, the method comprising The operation of an automatic programmed computing system in the performance of the following programmed steps: 